# Properties

 Label 8001.2.a.t Level $8001$ Weight $2$ Character orbit 8001.a Self dual yes Analytic conductor $63.888$ Analytic rank $0$ Dimension $16$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8001 = 3^{2} \cdot 7 \cdot 127$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8001.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$63.8883066572$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 2 x^{15} - 20 x^{14} + 38 x^{13} + 155 x^{12} - 275 x^{11} - 593 x^{10} + 957 x^{9} + 1177 x^{8} - 1655 x^{7} - 1150 x^{6} + 1279 x^{5} + 474 x^{4} - 280 x^{3} - 83 x^{2} + x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 889) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( 1 - \beta_{11} ) q^{5} - q^{7} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( 1 - \beta_{11} ) q^{5} - q^{7} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{8} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{10} - \beta_{11} + \beta_{12} + \beta_{15} ) q^{10} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} + \beta_{9} - \beta_{11} - \beta_{13} ) q^{11} + ( \beta_{2} - \beta_{3} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} ) q^{13} -\beta_{1} q^{14} + ( -1 - \beta_{2} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} + 2 \beta_{14} + \beta_{15} ) q^{16} + ( -\beta_{1} + \beta_{3} + \beta_{4} - \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + \beta_{14} ) q^{17} + ( -2 - \beta_{2} + \beta_{8} - \beta_{11} + 2 \beta_{14} ) q^{19} + ( 2 - \beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{13} + \beta_{14} ) q^{20} + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{22} + ( 3 - \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{11} - \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{23} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{13} - 2 \beta_{15} ) q^{25} + ( 1 + \beta_{1} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{14} + \beta_{15} ) q^{26} + ( -1 - \beta_{2} ) q^{28} + ( -1 - \beta_{2} + \beta_{5} - \beta_{6} + \beta_{8} + 2 \beta_{14} ) q^{29} + ( -1 - \beta_{1} - 4 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} + \beta_{9} - 2 \beta_{11} - \beta_{12} - 2 \beta_{13} - \beta_{15} ) q^{31} + ( -1 - \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{10} + \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{32} + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{10} + 2 \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{34} + ( -1 + \beta_{11} ) q^{35} + ( \beta_{1} + \beta_{3} + \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{13} ) q^{37} + ( 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{8} + \beta_{12} + \beta_{13} + \beta_{15} ) q^{38} + ( -3 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} - \beta_{10} - \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{40} + ( 3 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{13} - \beta_{15} ) q^{41} + ( 2 + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{8} - \beta_{9} - 2 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{43} + ( 1 + \beta_{1} + 3 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{44} + ( 4 + 4 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - 3 \beta_{12} - 3 \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{46} + ( 2 - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{15} ) q^{47} + q^{49} + ( -1 + 4 \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{6} - \beta_{9} - 4 \beta_{10} - \beta_{11} + 2 \beta_{12} - 2 \beta_{14} + 2 \beta_{15} ) q^{50} + ( 1 + 4 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} + 3 \beta_{15} ) q^{52} + ( 2 - 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{8} + 2 \beta_{10} + \beta_{11} - 2 \beta_{12} - 3 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{53} + ( 2 + \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{11} - \beta_{12} - \beta_{14} + \beta_{15} ) q^{55} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{56} + ( 1 + 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + \beta_{10} + 2 \beta_{11} + \beta_{12} + 2 \beta_{13} ) q^{58} + ( 5 + 2 \beta_{1} + 3 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{13} - 3 \beta_{14} - 2 \beta_{15} ) q^{59} + ( 4 - \beta_{1} + 6 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - \beta_{12} - 3 \beta_{13} - 4 \beta_{14} - \beta_{15} ) q^{61} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{10} + 2 \beta_{11} - 2 \beta_{14} ) q^{62} + ( -3 - 2 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} - \beta_{5} + 3 \beta_{8} - \beta_{9} + 3 \beta_{10} + 3 \beta_{11} + 3 \beta_{12} + 3 \beta_{13} + 2 \beta_{14} ) q^{64} + ( -3 + 4 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} - \beta_{9} - 3 \beta_{10} - \beta_{11} + 2 \beta_{12} + 3 \beta_{13} + 3 \beta_{15} ) q^{65} + ( 2 + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{14} - \beta_{15} ) q^{67} + ( 4 - 2 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} + 2 \beta_{13} + 2 \beta_{14} ) q^{68} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{10} + \beta_{11} - \beta_{12} - \beta_{15} ) q^{70} + ( 6 - \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{12} - 2 \beta_{13} ) q^{71} + ( -1 - \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} - 2 \beta_{9} + 3 \beta_{10} + \beta_{11} - \beta_{13} ) q^{73} + ( 4 + 3 \beta_{2} - 5 \beta_{3} - 3 \beta_{4} + \beta_{5} + 2 \beta_{6} - 3 \beta_{8} + \beta_{9} - 2 \beta_{10} - 3 \beta_{11} - 2 \beta_{12} - 3 \beta_{13} - 4 \beta_{14} - \beta_{15} ) q^{74} + ( \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{14} ) q^{76} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{9} + \beta_{11} + \beta_{13} ) q^{77} + ( -6 - 4 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{10} + 3 \beta_{12} + \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{79} + ( \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{8} - \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{80} + ( -4 + 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{8} - 2 \beta_{9} - 3 \beta_{10} + 3 \beta_{12} + \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{82} + ( 6 + 4 \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{6} + 3 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} + \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{83} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{8} + 2 \beta_{10} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{85} + ( 3 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{10} - 3 \beta_{11} - \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{86} + ( -4 + 6 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} + 6 \beta_{12} + 5 \beta_{13} + 5 \beta_{14} + 4 \beta_{15} ) q^{88} + ( 4 + 3 \beta_{1} + \beta_{2} - 5 \beta_{3} - 2 \beta_{4} - 3 \beta_{8} + \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{89} + ( -\beta_{2} + \beta_{3} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} ) q^{91} + ( 1 + 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - 4 \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} - 4 \beta_{14} ) q^{92} + ( 3 + \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{8} + \beta_{12} - 3 \beta_{14} - \beta_{15} ) q^{94} + ( -\beta_{4} + 2 \beta_{5} - \beta_{7} + 3 \beta_{8} - \beta_{9} + 3 \beta_{11} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{95} + ( 2 + 3 \beta_{2} + \beta_{5} + 2 \beta_{7} - 3 \beta_{8} - \beta_{11} - \beta_{13} - \beta_{14} ) q^{97} + \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 2q^{2} + 12q^{4} + 9q^{5} - 16q^{7} + 6q^{8} + O(q^{10})$$ $$16q + 2q^{2} + 12q^{4} + 9q^{5} - 16q^{7} + 6q^{8} - 2q^{10} + 22q^{11} - 4q^{13} - 2q^{14} + 12q^{16} + 18q^{17} - 15q^{19} + 40q^{20} - 11q^{22} + 5q^{23} + 15q^{25} + 24q^{26} - 12q^{28} + 12q^{29} - 32q^{31} + 9q^{32} - 14q^{34} - 9q^{35} - 2q^{37} - 3q^{38} - 14q^{40} + 45q^{41} - 3q^{43} + 54q^{44} + 49q^{47} + 16q^{49} + 6q^{50} + 38q^{52} - 16q^{53} + 7q^{55} - 6q^{56} + 16q^{58} + 35q^{59} - 11q^{61} - 17q^{62} - 2q^{64} - 14q^{65} + 17q^{67} + 71q^{68} + 2q^{70} + 81q^{71} - 15q^{73} - 13q^{74} + 14q^{76} - 22q^{77} - 34q^{79} + 33q^{80} - 14q^{82} + 39q^{83} - 17q^{85} - 36q^{86} + 61q^{88} + 32q^{89} + 4q^{91} - 37q^{92} + 13q^{94} + 33q^{95} - 4q^{97} + 2q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 2 x^{15} - 20 x^{14} + 38 x^{13} + 155 x^{12} - 275 x^{11} - 593 x^{10} + 957 x^{9} + 1177 x^{8} - 1655 x^{7} - 1150 x^{6} + 1279 x^{5} + 474 x^{4} - 280 x^{3} - 83 x^{2} + x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$($$$$-116093 \nu^{15} + 228893 \nu^{14} + 2347523 \nu^{13} - 4412026 \nu^{12} - 18385016 \nu^{11} + 32533154 \nu^{10} + 70696203 \nu^{9} - 115867538 \nu^{8} - 138531569 \nu^{7} + 205570931 \nu^{6} + 126910371 \nu^{5} - 162346281 \nu^{4} - 41913198 \nu^{3} + 34856907 \nu^{2} + 6759511 \nu - 128712$$$$)/239255$$ $$\beta_{4}$$ $$=$$ $$($$$$118542 \nu^{15} - 282272 \nu^{14} - 2211727 \nu^{13} + 5170999 \nu^{12} + 15614719 \nu^{11} - 35484771 \nu^{10} - 52971862 \nu^{9} + 114412902 \nu^{8} + 91508371 \nu^{7} - 178289744 \nu^{6} - 79215634 \nu^{5} + 121751459 \nu^{4} + 32149637 \nu^{3} - 26089138 \nu^{2} - 4109094 \nu + 299938$$$$)/239255$$ $$\beta_{5}$$ $$=$$ $$($$$$246643 \nu^{15} - 556793 \nu^{14} - 4734663 \nu^{13} + 10433381 \nu^{12} + 34632721 \nu^{11} - 74008394 \nu^{10} - 121920963 \nu^{9} + 250361358 \nu^{8} + 213453449 \nu^{7} - 416685796 \nu^{6} - 167476866 \nu^{5} + 306694906 \nu^{4} + 40444858 \nu^{3} - 64193652 \nu^{2} - 2974546 \nu + 1167877$$$$)/239255$$ $$\beta_{6}$$ $$=$$ $$($$$$-416133 \nu^{15} + 848713 \nu^{14} + 8205093 \nu^{13} - 16039026 \nu^{12} - 62183706 \nu^{11} + 115239599 \nu^{10} + 229254488 \nu^{9} - 397334578 \nu^{8} - 425687974 \nu^{7} + 679630526 \nu^{6} + 361459511 \nu^{5} - 519191331 \nu^{4} - 101394473 \nu^{3} + 112731547 \nu^{2} + 10770031 \nu - 1601912$$$$)/239255$$ $$\beta_{7}$$ $$=$$ $$($$$$-420868 \nu^{15} + 831558 \nu^{14} + 8486798 \nu^{13} - 16040641 \nu^{12} - 66156781 \nu^{11} + 118505764 \nu^{10} + 252322388 \nu^{9} - 423867643 \nu^{8} - 486918174 \nu^{7} + 758894461 \nu^{6} + 431479801 \nu^{5} - 611916381 \nu^{4} - 127789533 \nu^{3} + 142341487 \nu^{2} + 15316331 \nu - 2488852$$$$)/239255$$ $$\beta_{8}$$ $$=$$ $$($$$$86516 \nu^{15} - 193256 \nu^{14} - 1686213 \nu^{13} + 3671187 \nu^{12} + 12597346 \nu^{11} - 26550396 \nu^{10} - 45669957 \nu^{9} + 92287029 \nu^{8} + 83323677 \nu^{7} - 159344793 \nu^{6} - 69683347 \nu^{5} + 123008780 \nu^{4} + 19521756 \nu^{3} - 27300111 \nu^{2} - 2391162 \nu + 403609$$$$)/47851$$ $$\beta_{9}$$ $$=$$ $$($$$$-558146 \nu^{15} + 1242406 \nu^{14} + 10825416 \nu^{13} - 23508057 \nu^{12} - 80263942 \nu^{11} + 169127043 \nu^{10} + 287448331 \nu^{9} - 583976571 \nu^{8} - 513565088 \nu^{7} + 1000551217 \nu^{6} + 412351657 \nu^{5} - 766742622 \nu^{4} - 104612936 \nu^{3} + 169828364 \nu^{2} + 14520502 \nu - 2398759$$$$)/239255$$ $$\beta_{10}$$ $$=$$ $$($$$$609731 \nu^{15} - 1339991 \nu^{14} - 11975331 \nu^{13} + 25605932 \nu^{12} + 90323612 \nu^{11} - 186671853 \nu^{10} - 331094336 \nu^{9} + 655602681 \nu^{8} + 610664783 \nu^{7} - 1145738667 \nu^{6} - 514021097 \nu^{5} + 894448927 \nu^{4} + 142265856 \nu^{3} - 197861309 \nu^{2} - 18219637 \nu + 2939939$$$$)/239255$$ $$\beta_{11}$$ $$=$$ $$($$$$-614167 \nu^{15} + 1330387 \nu^{14} + 12063962 \nu^{13} - 25378599 \nu^{12} - 90934424 \nu^{11} + 184614846 \nu^{10} + 332488247 \nu^{9} - 646723747 \nu^{8} - 608767076 \nu^{7} + 1127337289 \nu^{6} + 501713704 \nu^{5} - 878849284 \nu^{4} - 127489287 \nu^{3} + 195153183 \nu^{2} + 14029589 \nu - 2942448$$$$)/239255$$ $$\beta_{12}$$ $$=$$ $$($$$$-673264 \nu^{15} + 1527129 \nu^{14} + 13105079 \nu^{13} - 29129163 \nu^{12} - 97629743 \nu^{11} + 211857962 \nu^{10} + 351724054 \nu^{9} - 742060049 \nu^{8} - 632322762 \nu^{7} + 1294215363 \nu^{6} + 508299188 \nu^{5} - 1013016703 \nu^{4} - 121901624 \nu^{3} + 232294846 \nu^{2} + 12764513 \nu - 5229541$$$$)/239255$$ $$\beta_{13}$$ $$=$$ $$($$$$907437 \nu^{15} - 1921932 \nu^{14} - 18024552 \nu^{13} + 36966909 \nu^{12} + 137780884 \nu^{11} - 271915646 \nu^{10} - 512234712 \nu^{9} + 966370827 \nu^{8} + 954511391 \nu^{7} - 1714237034 \nu^{6} - 797268019 \nu^{5} + 1362704444 \nu^{4} + 198561717 \nu^{3} - 308068303 \nu^{2} - 20676569 \nu + 5318248$$$$)/239255$$ $$\beta_{14}$$ $$=$$ $$($$$$-183898 \nu^{15} + 398886 \nu^{14} + 3621479 \nu^{13} - 7620756 \nu^{12} - 27413174 \nu^{11} + 55548928 \nu^{10} + 100969347 \nu^{9} - 195111036 \nu^{8} - 187405313 \nu^{7} + 341269962 \nu^{6} + 159092271 \nu^{5} - 267285935 \nu^{4} - 44412449 \nu^{3} + 59983859 \nu^{2} + 5138082 \nu - 1000302$$$$)/47851$$ $$\beta_{15}$$ $$=$$ $$($$$$929511 \nu^{15} - 2103041 \nu^{14} - 18058381 \nu^{13} + 39940442 \nu^{12} + 134317747 \nu^{11} - 288743733 \nu^{10} - 483981381 \nu^{9} + 1003060396 \nu^{8} + 875314583 \nu^{7} - 1730295122 \nu^{6} - 721664127 \nu^{5} + 1334031662 \nu^{4} + 195849826 \nu^{3} - 296868004 \nu^{2} - 24599592 \nu + 5430944$$$$)/239255$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 5 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{15} + 2 \beta_{14} - \beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} + 5 \beta_{2} + 13$$ $$\nu^{5}$$ $$=$$ $$9 \beta_{15} + 9 \beta_{14} + 10 \beta_{13} + 10 \beta_{12} + \beta_{11} + \beta_{10} - 9 \beta_{5} + 10 \beta_{4} + 12 \beta_{3} - 9 \beta_{2} + 28 \beta_{1} - 9$$ $$\nu^{6}$$ $$=$$ $$10 \beta_{15} + 22 \beta_{14} + 3 \beta_{13} + 3 \beta_{12} - 7 \beta_{11} + 3 \beta_{10} + 9 \beta_{9} + 13 \beta_{8} - 10 \beta_{7} - \beta_{5} + 2 \beta_{4} + 6 \beta_{3} + 24 \beta_{2} + 63$$ $$\nu^{7}$$ $$=$$ $$68 \beta_{15} + 72 \beta_{14} + 81 \beta_{13} + 81 \beta_{12} + 15 \beta_{11} + 16 \beta_{10} - \beta_{9} + 3 \beta_{8} - 3 \beta_{7} + \beta_{6} - 66 \beta_{5} + 80 \beta_{4} + 107 \beta_{3} - 70 \beta_{2} + 166 \beta_{1} - 71$$ $$\nu^{8}$$ $$=$$ $$81 \beta_{15} + 193 \beta_{14} + 49 \beta_{13} + 45 \beta_{12} - 30 \beta_{11} + 46 \beta_{10} + 64 \beta_{9} + 124 \beta_{8} - 81 \beta_{7} - 16 \beta_{5} + 35 \beta_{4} + 92 \beta_{3} + 109 \beta_{2} + \beta_{1} + 320$$ $$\nu^{9}$$ $$=$$ $$492 \beta_{15} + 555 \beta_{14} + 618 \beta_{13} + 614 \beta_{12} + 161 \beta_{11} + 171 \beta_{10} - 14 \beta_{9} + 49 \beta_{8} - 45 \beta_{7} + 12 \beta_{6} - 460 \beta_{5} + 599 \beta_{4} + 864 \beta_{3} - 517 \beta_{2} + 1021 \beta_{1} - 537$$ $$\nu^{10}$$ $$=$$ $$619 \beta_{15} + 1562 \beta_{14} + 537 \beta_{13} + 469 \beta_{12} - 45 \beta_{11} + 490 \beta_{10} + 428 \beta_{9} + 1041 \beta_{8} - 614 \beta_{7} - 2 \beta_{6} - 172 \beta_{5} + 398 \beta_{4} + 969 \beta_{3} + 448 \beta_{2} + 22 \beta_{1} + 1658$$ $$\nu^{11}$$ $$=$$ $$3511 \beta_{15} + 4202 \beta_{14} + 4602 \beta_{13} + 4524 \beta_{12} + 1495 \beta_{11} + 1567 \beta_{10} - 134 \beta_{9} + 549 \beta_{8} - 469 \beta_{7} + 96 \beta_{6} - 3169 \beta_{5} + 4376 \beta_{4} + 6668 \beta_{3} - 3728 \beta_{2} + 6447 \beta_{1} - 3961$$ $$\nu^{12}$$ $$=$$ $$4639 \beta_{15} + 12147 \beta_{14} + 5003 \beta_{13} + 4231 \beta_{12} + 822 \beta_{11} + 4497 \beta_{10} + 2816 \beta_{9} + 8185 \beta_{8} - 4524 \beta_{7} - 46 \beta_{6} - 1592 \beta_{5} + 3796 \beta_{4} + 8793 \beta_{3} + 1437 \beta_{2} + 322 \beta_{1} + 8626$$ $$\nu^{13}$$ $$=$$ $$24939 \beta_{15} + 31494 \beta_{14} + 33856 \beta_{13} + 32868 \beta_{12} + 12804 \beta_{11} + 13298 \beta_{10} - 1092 \beta_{9} + 5271 \beta_{8} - 4231 \beta_{7} + 631 \beta_{6} - 21848 \beta_{5} + 31652 \beta_{4} + 50232 \beta_{3} - 26569 \beta_{2} + 41544 \beta_{1} - 28730$$ $$\nu^{14}$$ $$=$$ $$34545 \beta_{15} + 92328 \beta_{14} + 42883 \beta_{13} + 35508 \beta_{12} + 13085 \beta_{11} + 38161 \beta_{10} + 18504 \beta_{9} + 62007 \beta_{8} - 32868 \beta_{7} - 653 \beta_{6} - 13716 \beta_{5} + 33079 \beta_{4} + 74013 \beta_{3} + 702 \beta_{2} + 3857 \beta_{1} + 44462$$ $$\nu^{15}$$ $$=$$ $$176958 \beta_{15} + 234588 \beta_{14} + 247371 \beta_{13} + 237044 \beta_{12} + 104286 \beta_{11} + 107926 \beta_{10} - 8127 \beta_{9} + 46678 \beta_{8} - 35508 \beta_{7} + 3571 \beta_{6} - 151318 \beta_{5} + 228008 \beta_{4} + 373091 \beta_{3} - 188317 \beta_{2} + 272149 \beta_{1} - 206072$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.57541 −2.24781 −1.72082 −1.48078 −1.45188 −0.532475 −0.191617 −0.170601 0.102239 0.625678 1.24549 1.33134 1.79993 2.18919 2.37299 2.70451
−2.57541 0 4.63271 3.61540 0 −1.00000 −6.78030 0 −9.31112
1.2 −2.24781 0 3.05263 0.411504 0 −1.00000 −2.36611 0 −0.924982
1.3 −1.72082 0 0.961212 4.11468 0 −1.00000 1.78756 0 −7.08062
1.4 −1.48078 0 0.192706 −1.52223 0 −1.00000 2.67620 0 2.25409
1.5 −1.45188 0 0.107956 −0.584615 0 −1.00000 2.74702 0 0.848791
1.6 −0.532475 0 −1.71647 0.118172 0 −1.00000 1.97893 0 −0.0629236
1.7 −0.191617 0 −1.96328 −3.92161 0 −1.00000 0.759431 0 0.751445
1.8 −0.170601 0 −1.97090 0.206353 0 −1.00000 0.677439 0 −0.0352041
1.9 0.102239 0 −1.98955 −0.280585 0 −1.00000 −0.407889 0 −0.0286868
1.10 0.625678 0 −1.60853 0.920707 0 −1.00000 −2.25778 0 0.576066
1.11 1.24549 0 −0.448744 2.43368 0 −1.00000 −3.04990 0 3.03114
1.12 1.33134 0 −0.227522 2.92411 0 −1.00000 −2.96560 0 3.89299
1.13 1.79993 0 1.23975 −3.74624 0 −1.00000 −1.36840 0 −6.74297
1.14 2.18919 0 2.79257 −0.868150 0 −1.00000 1.73509 0 −1.90055
1.15 2.37299 0 3.63108 3.84175 0 −1.00000 3.87053 0 9.11644
1.16 2.70451 0 5.31438 1.33706 0 −1.00000 8.96376 0 3.61609
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$1$$
$$127$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8001.2.a.t 16
3.b odd 2 1 889.2.a.c 16
21.c even 2 1 6223.2.a.k 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
889.2.a.c 16 3.b odd 2 1
6223.2.a.k 16 21.c even 2 1
8001.2.a.t 16 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8001))$$:

 $$T_{2}^{16} - \cdots$$ $$T_{5}^{16} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T - 83 T^{2} - 280 T^{3} + 474 T^{4} + 1279 T^{5} - 1150 T^{6} - 1655 T^{7} + 1177 T^{8} + 957 T^{9} - 593 T^{10} - 275 T^{11} + 155 T^{12} + 38 T^{13} - 20 T^{14} - 2 T^{15} + T^{16}$$
$3$ $$T^{16}$$
$5$ $$16 - 184 T + 92 T^{2} + 3406 T^{3} - 4663 T^{4} - 13233 T^{5} + 18066 T^{6} + 12824 T^{7} - 21504 T^{8} + 223 T^{9} + 7523 T^{10} - 2135 T^{11} - 532 T^{12} + 273 T^{13} - 7 T^{14} - 9 T^{15} + T^{16}$$
$7$ $$( 1 + T )^{16}$$
$11$ $$-417463 - 3453876 T - 4890377 T^{2} + 10632471 T^{3} + 8580746 T^{4} - 9620054 T^{5} - 2554535 T^{6} + 3416018 T^{7} - 78450 T^{8} - 494092 T^{9} + 96093 T^{10} + 21231 T^{11} - 8710 T^{12} + 563 T^{13} + 127 T^{14} - 22 T^{15} + T^{16}$$
$13$ $$9972315 + 44456121 T + 68885379 T^{2} + 34669992 T^{3} - 14684252 T^{4} - 18716919 T^{5} - 1195794 T^{6} + 3414846 T^{7} + 617394 T^{8} - 314125 T^{9} - 71893 T^{10} + 15670 T^{11} + 3891 T^{12} - 400 T^{13} - 101 T^{14} + 4 T^{15} + T^{16}$$
$17$ $$22881193 + 127487708 T - 189027016 T^{2} - 165982949 T^{3} + 400769219 T^{4} - 222118551 T^{5} + 14328251 T^{6} + 27320024 T^{7} - 8196701 T^{8} - 390984 T^{9} + 494484 T^{10} - 47312 T^{11} - 9260 T^{12} + 1809 T^{13} + T^{14} - 18 T^{15} + T^{16}$$
$19$ $$906809 - 25499416 T + 32453570 T^{2} + 33534422 T^{3} - 45556324 T^{4} - 16234226 T^{5} + 19332413 T^{6} + 4783381 T^{7} - 3183335 T^{8} - 769833 T^{9} + 204744 T^{10} + 59290 T^{11} - 3287 T^{12} - 1751 T^{13} - 56 T^{14} + 15 T^{15} + T^{16}$$
$23$ $$1853701360 + 748320776 T - 2004986780 T^{2} - 720099710 T^{3} + 727691803 T^{4} + 205682974 T^{5} - 122742602 T^{6} - 25351519 T^{7} + 10827289 T^{8} + 1571325 T^{9} - 524223 T^{10} - 50912 T^{11} + 13906 T^{12} + 814 T^{13} - 188 T^{14} - 5 T^{15} + T^{16}$$
$29$ $$8925976496 - 23320445808 T + 16209314744 T^{2} + 295371266 T^{3} - 3929648815 T^{4} + 835790244 T^{5} + 324777535 T^{6} - 113342531 T^{7} - 10104526 T^{8} + 6387491 T^{9} - 31205 T^{10} - 177735 T^{11} + 8524 T^{12} + 2378 T^{13} - 168 T^{14} - 12 T^{15} + T^{16}$$
$31$ $$713813 + 43245292 T - 36879201 T^{2} - 1166623988 T^{3} - 332307795 T^{4} + 428282619 T^{5} + 131411798 T^{6} - 53539080 T^{7} - 18108144 T^{8} + 2578482 T^{9} + 1119579 T^{10} - 16797 T^{11} - 30548 T^{12} - 1864 T^{13} + 248 T^{14} + 32 T^{15} + T^{16}$$
$37$ $$-5726959039 + 35513683307 T - 54649415316 T^{2} - 6162548967 T^{3} + 18948149064 T^{4} - 599680907 T^{5} - 2023401835 T^{6} + 91517416 T^{7} + 102779604 T^{8} - 3926772 T^{9} - 2819583 T^{10} + 75206 T^{11} + 42522 T^{12} - 650 T^{13} - 328 T^{14} + 2 T^{15} + T^{16}$$
$41$ $$19825104281 - 66630794266 T + 76024909115 T^{2} - 29479897342 T^{3} - 5523628007 T^{4} + 7585034668 T^{5} - 1692477753 T^{6} - 241669271 T^{7} + 166915016 T^{8} - 22266705 T^{9} - 1615297 T^{10} + 738376 T^{11} - 62811 T^{12} - 2067 T^{13} + 698 T^{14} - 45 T^{15} + T^{16}$$
$43$ $$-219978965872 + 223969148376 T + 12096314108 T^{2} - 73182935122 T^{3} + 13360158699 T^{4} + 7462939342 T^{5} - 2080833947 T^{6} - 319901908 T^{7} + 124470758 T^{8} + 5409270 T^{9} - 3609001 T^{10} - 6953 T^{11} + 52518 T^{12} - 522 T^{13} - 369 T^{14} + 3 T^{15} + T^{16}$$
$47$ $$49063403511 - 40419022167 T - 72572388210 T^{2} + 14910084996 T^{3} + 22445732749 T^{4} - 4122522034 T^{5} - 2706946462 T^{6} + 658523530 T^{7} + 103910175 T^{8} - 40586699 T^{9} + 780789 T^{10} + 856580 T^{11} - 92898 T^{12} - 1387 T^{13} + 788 T^{14} - 49 T^{15} + T^{16}$$
$53$ $$114032072912 + 383764785576 T + 385717125760 T^{2} + 72376451796 T^{3} - 72211379599 T^{4} - 26882054279 T^{5} + 3784318232 T^{6} + 2203921994 T^{7} - 26555696 T^{8} - 71660412 T^{9} - 2164870 T^{10} + 1058843 T^{11} + 51113 T^{12} - 6958 T^{13} - 400 T^{14} + 16 T^{15} + T^{16}$$
$59$ $$-58448663216 - 137797908472 T + 345017393284 T^{2} + 125110417062 T^{3} - 112428613345 T^{4} - 18710440916 T^{5} + 13073423507 T^{6} + 333908077 T^{7} - 577745292 T^{8} + 21769814 T^{9} + 10929012 T^{10} - 822205 T^{11} - 81931 T^{12} + 9459 T^{13} + 77 T^{14} - 35 T^{15} + T^{16}$$
$61$ $$-123057095311 + 128263909368 T + 698618299855 T^{2} - 1107711472685 T^{3} + 405214040969 T^{4} + 22323340845 T^{5} - 31128166128 T^{6} + 1699324278 T^{7} + 905912369 T^{8} - 67774300 T^{9} - 13951967 T^{10} + 927525 T^{11} + 122146 T^{12} - 5360 T^{13} - 557 T^{14} + 11 T^{15} + T^{16}$$
$67$ $$15509381648 + 35007259120 T - 46707500984 T^{2} - 59085181934 T^{3} + 6000825861 T^{4} + 11351514590 T^{5} - 207075423 T^{6} - 880272930 T^{7} + 4759078 T^{8} + 33866686 T^{9} - 510151 T^{10} - 668507 T^{11} + 21819 T^{12} + 6073 T^{13} - 301 T^{14} - 17 T^{15} + T^{16}$$
$71$ $$120439326815165 - 86909851091416 T - 8001665506318 T^{2} + 25881402447621 T^{3} - 9556953355470 T^{4} + 1089088776512 T^{5} + 173004023146 T^{6} - 66341816167 T^{7} + 6475793641 T^{8} + 251785774 T^{9} - 112937112 T^{10} + 9687622 T^{11} - 156864 T^{12} - 31055 T^{13} + 2524 T^{14} - 81 T^{15} + T^{16}$$
$73$ $$-145782520397 + 126181266892 T + 1173459269368 T^{2} + 866120413554 T^{3} - 58046867196 T^{4} - 161682712504 T^{5} - 11710717518 T^{6} + 8587055612 T^{7} + 594844521 T^{8} - 186687435 T^{9} - 12085138 T^{10} + 1915022 T^{11} + 122342 T^{12} - 9018 T^{13} - 586 T^{14} + 15 T^{15} + T^{16}$$
$79$ $$845588972425 - 100426808457141 T + 26292877959958 T^{2} + 18455256702068 T^{3} - 3199910499259 T^{4} - 1120645248693 T^{5} + 132365790057 T^{6} + 32422541979 T^{7} - 2335430200 T^{8} - 508545617 T^{9} + 17369948 T^{10} + 4389858 T^{11} - 23912 T^{12} - 19389 T^{13} - 292 T^{14} + 34 T^{15} + T^{16}$$
$83$ $$-3653038864 - 40796521408 T - 31984608144 T^{2} + 106004834126 T^{3} - 1959097315 T^{4} - 42205527522 T^{5} + 4635252383 T^{6} + 3822765984 T^{7} - 728396318 T^{8} - 61632483 T^{9} + 21893349 T^{10} - 696166 T^{11} - 173325 T^{12} + 13847 T^{13} + 103 T^{14} - 39 T^{15} + T^{16}$$
$89$ $$10200551550832 - 25509130372024 T + 5224541800720 T^{2} + 6569852274968 T^{3} - 2267052626127 T^{4} - 279092882329 T^{5} + 174848212292 T^{6} - 7358226445 T^{7} - 3781844514 T^{8} + 351277874 T^{9} + 29078299 T^{10} - 4163337 T^{11} - 45150 T^{12} + 19583 T^{13} - 336 T^{14} - 32 T^{15} + T^{16}$$
$97$ $$-269740302224 + 51941075160 T + 376551165772 T^{2} - 141593628910 T^{3} - 120348358705 T^{4} + 76399136638 T^{5} - 6119655493 T^{6} - 3968609010 T^{7} + 716406141 T^{8} + 59307316 T^{9} - 18000751 T^{10} - 100598 T^{11} + 169570 T^{12} - 1511 T^{13} - 697 T^{14} + 4 T^{15} + T^{16}$$